Chordal embeddings of planar graphs - LaBRI

**Chordal** **embeddings** **of** **planar** **graphs**

V. Bouchitté 1 , F. Mazoit 1 , and I. Todinca 2

1 LIP-École Normale Supérieure de Lyon

46 Allée d’Italie, 69364 Lyon Cedex 07, France

email: {Vincent.Bouchitte,Frederic.Mazoit}@ens-lyon.fr

2 LIFO-Université d’Orléans

BP 6759, 45067 Orleans Cedex 2, France

email: Ioan.Todinca@lifo.univ-orleans.fr

Abstract. Robertson and Seymour conjectured that the treewidth **of** a **planar**

graph and the treewidth **of** its geometric dual differ by at most one.

Lapoire solved the conjecture in the affirmative, using algebraic techniques.

We give here a much shorter pro**of** **of** this result.

1 Introduction

The notions **of** treewidth and tree decomposition **of** a graph have been introduced by

Robertson and Seymour in [14] for their study **of** minors **of** **graphs**. These notions

have been intensively investigated for algorithmical purposes since many NP-hard

problems become polynomial and even linear when restricted to classes **of** **graphs**

with bounded treewidth.

Robertson and Seymour conjectured in [13] that the treewidth **of** a **planar** graph

and the treewidth **of** its geometric dual differ by at most one. Lapoire [11] solved

this conjecture in the affirmative, in fact he proved a more general result. In order

to prove his result, Lapoire worked on hypermaps and introduced the notion **of**

splitting **of** hypermaps, his approach is essentially an algebraic one.

Computing the treewidth **of** an arbitrary graph is NP-hard. Nevertheless, the

treewidth can be computed in polynomial time for several well-known classes **of**

**graphs**, for example chordal bipartite **graphs** [9], circle and circular-arc **graphs** [8]

[16], permutation **graphs** [2] and weakly triangulated **graphs** [3]. Actually all these

classes **of** **graphs** have a polynomial number **of** minimal separators, we proved in

[4] that we can compute, in polynomial time, the treewidth **of** a graph in any class

having a polynomial number **of** minimal separators.

For classes **of** **graphs** having an exponential number **of** minimal separators, we

know very few, for instance the problem remains NP-hard on AT-free **graphs** [1] and

it is polynomial for rectangular grids. Maybe the most challenging open problem is

the computation **of** the treewidth for **planar** **graphs**. In [15], Seymour and Thomas

gave a polynomial time algorithm that approximate the treewidth **of** **planar** graph

within a factor **of** 3 2 .

In this paper, we give a new approach to tackle the problem **of** the treewidth

computation for **planar** **graphs**. First, we recall how to obtain minimal chordal

**embeddings** **of** **graphs** by completing some families **of** minimal separators. Secondly,

we show that we can interpret minimal separators **of** **planar** **graphs** as Jordan curves

**of** the plane. Then, we study the structure **of** Jordan curves that give a minimal

triangulation **of** the graph. Next, given a family **of** curves **of** the plane, we show how

to build a minimal triangulation **of** the geometric dual **of** the graph. Finally, given

an optimal triangulation w.r.t. treewidth **of** the initial graph, we give a triangulation

**of** the dual graph whose maximal cliquesize is no more than the maximal cliquesize

**of** the original graph plus one. So, we get a new pro**of** **of** the conjecture **of** Robertson

and Seymour which is much simpler than the pro**of** **of** Lapoire.

2 Preliminaries

Throughout this paper we consider simple, finite, undirected **graphs**.

A graphG=(V,E) is **planar** if it can be drawn in the plane such that no two

edges meet in a point other than a common end. The plane will be denoted byΣ. A

plane graphG = (V,E) is a drawing **of** a **planar** graph. That is, each vertexv∈V

is a point **of**Σ, each edgee∈E is a curve between two vertices, distinct edges have

distinct sets **of** endpoints and the interior **of** an edge contains no point **of** another

edge. A face **of** the plane graphG is a region **of**Σ\G.F(G) denotes the set **of** faces

**of**G. Sometimes we will also use plane multi**graphs**, i.e. we allow loops and multiple

edges.

LetG=(V,E) a plane graph. The dualG ∗ = (F,E ∗ ) **of**Gis a plane multigraph

obtained in the following way: for each face **of**G, we place a pointf into the face,

and these points form the vertex set **of**G ∗ . For each edgee**of**G, we link the two

vertices **of**G ∗ corresponding to faces incident toeinG, by an edgee ∗ crossinge; if

e is incident with only one face, thene ∗ is a loop.

A graphH is chordal (or triangulated) if every cycle **of** length at least four has

a chord. A triangulation **of** a graphG=(V,E) is a chordal graphH = (V,E ′ )

such thatE⊆E ′ .H is a minimal triangulation if for any intermediate setE ′′ with

E⊆E ′′ ⊂E ′ , the graph (V,E ′′ ) is not triangulated. We point out that in this

paper, a triangulation **of** a **planar** graphGwill always mean a chordal embedding

**of**G. Thus, a triangulation **of**Gis clearly not equivalent to a **planar** triangulation

(that is, a **planar** supergraph such that each face **of** the supergraph is a triangle) **of**

G.

Definition 1. LetG = (V,E) be a graph. The treewidth **of**G, denoted bytw(G),

is the minimum, over all triangulationsH **of**G, **of**ω(H)−1, whereω(H) is the

the maximum cliquesize **of**H. The treewidth **of** a multigraph is the treewidth **of** the

corresponding simple graph.

The aim **of** this paper is to prove the following assertion, stated by Robertson

and Seymour in [13]:

Problem 1. For any plane graphG = (V,E),

tw(G ∗ )≤tw(G)≤tw(G ∗ ) + 1.

We say that a graphG ′ is a minor **of** a graphG if we can obtainG ′ fromGby

repeatedly using the following operations: vertex deletion, edge deletion and edge

contraction. Kuratowski’s theorem states that a graphGis **planar** if and only if the

**graphs**K 3,3 andK 5 are not minors **of**G. It is well-known that ifG ′ is a minor **of**

G, then tw(G ′ )≤tw(G). We reffer to [5] for more details on these results.

When we compute the treewidth **of** a graphG, we are searching for a triangulation

**of**Gwith smallest cliquesize, so we can restrict our work to minimal triangulations.

We need a characterization **of** the minimal triangulations **of** a graph, using

the notion **of** minimal separator.

A subsetS⊆V is ana,b-separator for two nonadjacent verticesa,b∈V if the

removal **of**S from the graph separatesaandbin different connected components.

S is a minimala,b-separator if no proper subset **of**S separatesaandb. We say

thatS is a minimal separator **of**Gif there are two verticesaandbsuch thatS is

a minimala,b-separator. Notice that a minimal separator can be strictly included

into another. We denote by∆ G the set **of** all minimal separators **of**G.

LetGbe a graph andS be a minimal separator **of**G. We denote byC G (S) the

set **of** connected components **of**G\S. A componentC∈C G (S) is full if every vertex

**of**S is adjacent to some vertex **of**C. For the following lemma, we refer to [7].

2

Lemma 1. A setS **of** vertices **of**Gis a minimala,b-separator if and only ifaand

b are in different full components **of**S.

Definition 2. Two separatorsS andT cross, denoted byS♯T , if there are some

distinct componentsC andD**of**G\T such thatS intersects both **of** them. IfS and

T do not cross, they are called parallel, denoted byS‖T .

It is easy to prove that these relations are symmetric.

LetS∈∆ G be a minimal separator. We denote byG S the graph obtained

fromGby completingS, i.e. by adding an edge between every pair **of** non-adjacent

vertices **of**S. IfΓ⊆∆ G is a set **of** separators **of**G,G Γ is the graph obtained by

completing all the separators **of**Γ . The results **of** [10], concluded in [12], establish

a strong relation between the minimal triangulations **of** a graph and its minimal

separators.

Theorem 1. LetΓ∈∆ G be a maximal set **of** pairwise parallel separators **of**G.

ThenH=G Γ is a minimal triangulation **of**Gand∆ H =Γ .

LetH be a minimal triangulation **of** a graphG. Then∆ H is a maximal set **of**

pairwise parallel separators **of**GandH =G ∆H . Moreover, for eachS∈∆ H , the

connected components **of**H\S are exactly the connected components **of**G\S.

In other terms, every minimal triangulation **of** a graphGis obtained by considering

a maximal setΓ **of** pairwise parallel separators **of**Gand completing the

separators **of**Γ . The minimal separators **of** the triangulation are exactly the elements

**of**Γ .

3 Minimal separators as curves

We show in this section that, in plane **graphs**, we can associate to each minimal

separatorS a Jordan curve such that, ifS separates two vertices **of** the graph, then

the curve separates the corresponding points in the plane.

Definition 3. LetG = (V,E) be a **planar** graph. We fix a plane embedding **of**G.

LetF be the set **of** faces **of** this embedding. The intermediate graphG I = (V∪F,E I )

has vertex setV∪F. We place an edge inG I between an original vertexv∈V and

a face-vertexf∈F whenever the corresponding vertex and face are incident inG.

Proposition 1. LetGbe a 2-connected plane graph. Then the intermediate graph

G I is also 2-connected.

Pro**of**. Let us prove that, for any couple **of** original verticesxandy **of**G I and for

any face or original vertexa, there is anx,y-path inG I avoidinga. Letµ=[x =

v 1 ,v 2 ,...,v p =y] anx,y-path **of**G. Ifa∈V (G), since{a} is not anx,y separator

**of**G, we can chooseµsuch thata∉µ. For each edgee i =v i ,v i+1 , 1≤i

Proposition 2. Consider a cycleν **of**G I . Its drawing defines a Jordan curve ˜ν in

the plane. Removing ˜ν separates the plane into two regions. If both regions contain

at least one original vertex, then the original vertices **of**ν form a separator **of**G.

Pro**of**. Letxandybe two original vertices, separated by the curve ˜ν in the plane.

Clearly, no edge **of**Gcrosses an edge **of**G I , and therefore no edge **of**Gcrosses the

curve ˜ν. Every pathµconnectingxandy inGintersects ˜ν, soµhas a vertex inν.

It follows thatν∩V is ax,y-separator **of**G. ⊓⊔

Proposition 3. LetS be a minimal separator **of** a 2-connected plane graphGand

C be a full component associated toS. ThenScorresponds to an elementary cycle

ν S (C) **of**G I , **of** the same original vertices and **of** equal number **of** face-vertices in

G I , such thatG I \ν S (C) has at least two connected components. Moreover, the

original vertices **of** one **of** these components are exactly the vertices **of**C.

Pro**of**. LetC be a full component associated toS, letG C be formed by contracting

C into a supervertex, and letS ′ be the set **of** faces and vertices adjacent inG C to

the contracted supervertex. ThenS ′ is neighborhood **of** the supervertex inG C I , so

it has the structure **of** a cycle inG C I and therefore inG I. This cycle will be denoted

ν S (C). SinceC is a full component associated toS inG, we have thatS=N G (C),

so the original vertices **of**S ′ are exactly vertices **of**S. The cycle separatesC from

V\{S∪C} inG I .

⊓⊔

The cycleν S (C) defined in the previous proposition will be called the cycle

associated toS andC, close toC.

Remark 1. Any cycleν **of**G I forms a Jordan curve in the plane. We denote ˜ν this

curve. Removing ˜ν separates the plane into two open regions. Consider the cycle

ν S (C) **of**G I associated to a minimal separatorS and a full componentC **of**G\S,

close toC. Then one **of** the regions defined by ˜ν S (C) contains all the vertices **of**C

and the other contains all the vertices **of**V\ (S∪C).

4 Some technical lemmas

In the next section we show how to associate to each minimal separatorS **of** the

3-connected plane graphGaunique cycle **of**G I having good separation properties.

We group here some technical lemmas that will be used in the next sections.

Lemma 2. LetGbe a 3-connected **planar** graph andS be a minimal separator **of**

G. ThenG\S has exactly two connected components.

Pro**of**. By Lemma 1, there are two distinct full componentsC 1 andC 2 associated to

S. Suppose there is another componentC 3 **of**G\S and letS 3 =N(C 3 ). Clearly,S 3 is

a separator **of**G, so|S 3 |≥3. Letx 1 ,x 2 ,x 3 be three distinct vertices **of**S 3 . Consider

the plane graphG ′ obtained fromGby contracting each componentC 1 ,C 2 andC 3

into a supervertex. The three supervertices are adjacent inG ′ tox 1 ,x 2 ,x 3 , soG ′

contains a subgraph isomorphic toK 3,3 – contradicting Kuratowski’s theorem. ⊓⊔

Proposition 4. LetS be a minimal separator **of** a 3-connected **planar** graphG.

ThenS is also an inclusion minimal separator **of**G.

Pro**of**. Suppose there is a separatorT **of**Gsuch thatT⊂S. There is a connected

componentC **of**G\T such thatC∩S =∅. Indeed, ifS intersects each component

**of**G\T , thenS andT cross, and since the crossing relation is symmetricT must

intersect two connected components **of**G\S, contradictingT⊂S. SinceS∩C =∅

andT ⊂S,C is also a connected component **of**G\S. By Lemma 1, there are

two full componentsD 1 ,D 2 associated toS. Notice thatC is not a full component

associated toS, becauseN(C) =T⊂S. It follows thatD 1 ,D 2 andC are three

distinct components associated toS inG, contradicting Lemma 2. ⊓⊔

4

Lemma 3. LetGbe a plane graph andν be a cycle **of**G I such that ˜ν separates two

original verticesaandbin the plane. Consider two verticesxandy**of**ν. Suppose

there is a pathµfromxtoy inG I , such thata,b∉µ andµdoes not intersect the

cycleν except inxandy.

The verticesxandy splitν into twox,y-paths **of**G I , denotedµ 1 andµ 2 . Consider

the cyclesν 1 (respectivelyν 2 ) **of**G i formed by the pathsµandµ 1 (respectively

µ andµ 2 ). Then ˜ν 1 or ˜ν 2 separate two original vertices in the plane.

Pro**of**. LetR 1 ,R 2 be the two regions obtained by removing ˜ν from the plane. By

hypothesis, bothR 1 andR 2 contain original vertices, saya∈R 1 andb∈R 2 .

Suppose w.l.o.g. that the pathµis contained inR 1 ∪{x,y}. Then the drawing **of**µ

splitsR 1 into two regions:R 1 ′ , bordered by the curve ˜ν 1, andR 1 ′′,

bordered by ˜ν 2.

Ifa∈R 1 ′ then ˜ν 1 separatesaandbin the plane, otherwisea∈R 1 ′′ so ˜ν 2 separates

a andbin the plane.

⊓⊔

Lemma 4. LetS be a minimal separator **of** a 3-connected plane graphG. Consider

a cyleν S **of**G I such that the original vertices **of**ν S are the elements **of**S. Suppose

that ˜ν S separates in the plane two original vertices **of**G.

If two original vertices **of**S are at distance two inG I (i.e. they are incident to

a same face **of**G), these vertices are also at distance two on the cycleν S .

Pro**of**. Letν S = [v 1 ,f 1 ,...,v p ,f p ], wherev i (respectivelyf i ) are the original (respectively

face) vertices **of**ν S . The conclusion is obvious ifp≤3. Suppose there

are two verticesx,y∈S at distance two inG I , but not inν S . W.l.o.g., we suppose

x =v 1 andy=v i , 3≤i≤p−2. Letf be a face vertex adjacent tov 1 andv i in

G I .

Iff ∉ν S (C), we apply Lemma 3 with cycleν S and path [v 1 ,f,v i ], so one

**of** the cyclesν 1 = [v 1 ,f 1 ,v 2 ,f 2 ,...,v i ,f] orν 2 = [v 1 ,f,v i ,f i ,v i+1 ,f i+1 ,...,v p ,f p ]

separates two original vertices in the plane (see figure 1a). By proposition 2, the

original vertices **of**ν 1 orν 2 form a separatorT inG. ButT is strictly contained in

S, contradicting proposition 4.

v 1 f 1

f 1

v 2

v 2

v 1

f

f

v i+1 f i+1

˜ν S

f i v

v ˜ν S

i+1 i f i

(a)

(b)

Fig. 1. Pro**of** **of** Lemma 4

The casef∈ν S is very similar. There is somej, 1≤j≤psuch thatf =f j .

Sincev 1 andv i are not at distance two onν S , we have thatj∉{1,p} orj∉

{i−1,i + 1}. Suppose w.l.o.g. thatf is not consecutive tov 1 on the cycleν S . We

apply Lemma 3 with cycleν S and path [v 1 ,f]. We obtain that one **of** the cycles

ν 1 = [v 1 ,f 1 ,...,v j ,f j ] orν 2 = [v 1 ,f j ,v j+1 ,f j+1 ,...,v p ,f p ] (see figure 1b) separates

two original vertices **of**G, so the original vertices **of**ν 1 orν 2 form a separatorT **of**

G. In both cases,T⊂S, contradicting proposition 4. ⊓⊔

Lemma 5. LetG = (V,E) be a plane graph andx,y∈V such that at least three

faces are incident to bothxandy. Then{x,y} is a separator **of**G.

5

Pro**of**. Letf 1 ,f 2 ,f 3 be three faces incident to bothxandy. Consider the three

pathsµ i = [x,f i ,y], 1≤i≤3 **of** the intermediate graphG I . The drawings **of** these

paths split the plane into three regions:R 1 bordered by the cycleν 1 = [x,f 2 ,y,f 3 ],

R 2 bordered byν 2 = [x,f 1 ,y,f 3 ] andR 3 bordered byν 3 = [x,f 1 ,y,f 2 ]. We show

that at least two **of** the three regions contain one or more original vertices.

Suppose thatR 2 andR 3 do not contain original vertices. In the graphG, each

face is incident to at least three vertices, s**of** 1 has a neighborz, different fromx

andy. The edgef 1 z **of**G i does not cross any **of** the pathsµ 1 ,µ 2 ,µ 3 , soz is in one

**of** the regionsR 2 orR 3 , incident t**of** 1 . This contradicts our assumption thatR 2

andR 3 do not contain original vertices.

We proved that at least two **of** the three regionsR 1 ,R 2 ,R 3 – sayR 1 andR 2

– contain original vertices. Then ˜ν 1 , the Jordan curve borderingR 1 , separates two

original vertices **of**G. By proposition 2, the original vertices **of**µ 1 , namely{x,y},

form a separator **of**G.

⊓⊔

Lemma 6. LetG = (V,E) be a 3-connected plane graph andx,y∈V .

1. Ifxy∈E, there are exactly two faces incident to bothxandy.

2. Ifxy∉E, there is at most one face **of**Gincident to bothxandy.

Pro**of**. The graphGis 3-connected, so by Lemma 5 there are at most two faces

incident to bothxandy.

The first statement is obvious since the edgexy is incident to two faces **of**G.

For the second statement, suppose there are two facesf 1 andf 2 incident toxandy.

Consider the plane graphG ′ obtained fromGby adding the edgexy, drawn in the

facef 1 . Then the facef 1 **of**Gis splitted into two facesf 1 ′ andf′′ 1 , both incident to

x andy inG ′ . So, inG ′ , the three facesf 1 ′,f′′

1 andf 2 are incident to bothxandy.

ButG ′ is clearly a 3-connected **planar** graph, and by Lemma 5 we have that{x,y}

is a separator **of**G ′ – a contradiction. ⊓⊔

Proposition 5. LetGbe a 3-connected plane graph. Consider two cyclesν andν ′

**of**G I , such thatν andν ′ only differ by their face vertices. Then ˜ν separates two

original verticesaandbin the plane if and only if ˜ν ′ also separatesaandbin the

plane.

Pro**of**. It is sufficient to prove our statement for two cycles that only differ by one

face vertex, sayν = [v 1 ,f 1 ,v 2 ,f 2 ,...,v p ,f p ] andν ′ = [v 1 ,f 1,v ′ 2 ,f 2 ,...,v p ,f p ], such

thatf 1 ≠f 1 ′. Sincev 1 andv 2 are incident to bothf 1 andf 1 ′ inG, it comes by

Lemma 6 thatv 1 andv 2 are adjacent inG. Thus,f 1 andf 1 ′ are the faces incident

inGto the edgee =v 1 v 2 .

Consider the cycleν ′′ = [v 1 ,f 1 ,v 2 ,f 1 ′] **of**G I and letRbe the region bordered

by ˜ν ′′ and containing the interior **of** the edgee. Clearly, the regionRcontains no

original or face vertex **of**G I .

LetR 1 ,R 2 be the two regions obtained by removing ˜ν from the plane. Suppose

that the edgee, and thus the facef 1, ′ is inR 2 . Then the regions obtained by removing

˜ν ′ from the plane are exactlyR 1 ′ =R 1∪R∪[v 1 ,f 1 ,v 2 ] andR 2 ′ =R 2\R\[v 1 ,f 1 ′,v 2].

SinceRcontains no original vertices, the original vertices **of**R 1 ′ (respectivelyR 2)

′

are the original vertices **of**R 1 (respectivelyR 2 ). ⊓⊔

Lemma 7 ([5], proposition 4.2.10). LetGbe a 3-connected plane graph. For

any facef, the set **of** vertices incident t**of** do not form a separator **of**G.

Lemma 8. LetGbe a 3-connected plane graph andS be a minimal separator **of**G.

Then each face **of**Gis incident to at most two vertices **of**S.

6

Pro**of**. Suppose there are three verticesx,y,z **of**S incident to a same facef. Let

C be a full component associated toSinGandν S (C) be the cycle associated

toSinG I , close toC. Consider first the case|S|≥4, so there is some vertex

t∈S\{x,y,z}. Suppose w.l.o.g. thatν S (C) encountersx,y,z andtin this order.

Thenxandz are not at distance 2 on the cycleν S (C), contradicting Lemma 4.

If|S| = 3, letT be the set **of** vertices incident t**of**, soS⊆T . Thus,T is a

separator **of**G, contradicting Lemma 7. ⊓⊔

5 Minimal separators in 3-connected **planar** **graphs**

Consider a minimal separatorS **of**Gand two full componentsC andDassociated

toS. We can associate toS two cycles **of**G I , namelyν S (C) andν S (D), closed to

C, respectivelyD. In general, the two cycles are distinct, although they represent

for us the same minimal separatorS. In the case **of** 3-connected **planar** **graphs**, we

slightly modify the construction **of** proposition 3 in order to obtain a unique cycle

representingS inG I .

LetGbe a 3-connected plane graph. Consider two original verticesxandy

situated at distance two inG I . We know thatxany are incident to a same face

inG, but this face is not necessarily unique. For each pair **of** verticesx,y∈V at

distance two inG I , we fix a unique facef(x,y) **of**Gincident to bothxandy. Let

ν = [v i ,f 1 ,v 2 ,f 2 ,...,v p ,f p ] be a cycle **of**G I , wherev i are the original vertices and

f i are the face vertices **of**ν. We say that a cycleν is well-formed if, for each pair

**of** consecutive original verticesv i ,v i+1 **of**ν we havef i =f(v i ,v i+1 ) (1≤i≤p,

v p+1 =v 1 ).

Given a minimal separatorS **of**Gwe construct a unique well-formed cycleν S

associated toS as follows.

LetC,D be the full components associateds toS inGand letν S (C) = [v 1 ,f 1 ,v 2 ,

f 2 ,...,v p ,f p ] be the cycle associated toS inG I , close toC. We denoteν S ′ (C) =

[v 1 ,f 1 ′,v 2,f 2 ′,...,v p,f p ′] wheref′ i =f(v i,v i+1 )∀i, 1≤i≤p. Notice that∀i,j, 1≤

i

We prove that ifν S crossesν T , thenS crossesT . LetRandR ′ be the regions

**of**Σ\˜ν S . We show that at least one original vertex **of**ν T is inR. Sinceν S crosses

ν T , ˜ν T intersectsR. Thus, an original vertex or a face-vertex **of**ν T is inR. Suppose

thatν T has no original vertex inRand letf be a face-vertex **of**ν T ∩R. On the

cycleν T , the face-vertexf is between two original verticesxandx ′ . Notice thatx

is also a vertex **of**ν S . Indeed,x∉R, andxcannot be inR ′ , because the edgexf

**of**G I cannot cross the drawing **of** the cycleν S . It follows thatx∈ ˜ν S . Soxand

x ′ are both vertices **of**ν S . Sincexandx ′ are adjacent to a same face-vertex **of**G I ,

they are on a same face **of**G. By Lemma 4,xandx ′ are at distance two on the

cycleν S , and letf ′ be the face-vertex **of**ν S betweenxandx ′ . Sinceν S andν T are

well-formed cycles, we havef ′ =f(x,y) =f, s**of**∈ν S . This contradicts the fact

thatf is in one **of** the regions **of**Σ\˜ν S .

We showed thatν S has original vertices in regionR, and for similar reasons it has

original vertices inR ′ . So ˜ν S separates two original vertices **of**ν T in the plane, and

by proposition 6S separates these vertices inG. Thus,S crossesT . ⊓⊔

6 Block regions

Let ˜ν be a Jordan curve in the plane. LetRbe one **of** the regions **of**Σ\˜ν. We say

that (˜ν,R) = ˜ν∪R is a one-block region **of** the plane, bordered by ˜ν.

Definition 5. Let ˜C be a set **of** curves such that for each ˜ν∈ ˜C, there is a oneblock

region (˜ν,R(˜ν)) containg all the curves **of** ˜C. We define the region between

the elements **of** ˜C as

RegBetween( ˜C) =

⋂˜ν∈ ˜C(˜ν,R(˜ν))

We say that the region between the curves **of** ˜C is bordered by ˜C.

Definition 6. A subsetBR⊆Σ **of** the plane is a block region if one **of** the following

holds:

–BR =Σ.

– There is a curve ˜ν such thatBR is a one-block region (˜ν,R).

– There is a set **of** curves ˜C such thatBR =RegBetween( ˜C).

Remark 2. According to our definition, block regions are always closed sets.

˜µ 2

˜µ 1 ˜µ 3

(a)

(b)

Fig. 2. Block regions

8

Example 1. In figure 2a, we have a block region (in grey) bordered by four Jordan

curves. Figure 2b presents three interior-disjoint paths ˜µ 1 , ˜µ 2 and ˜µ 3 having the

same endpoints. Consider the three curves ˜ν 1 = ˜µ 2 ∪˜µ 3 , ˜ν 2 = ˜µ 1 ∪˜µ 3 and ˜ν 3 = ˜µ 1 ∪˜µ 2 .

Notice that the block-region between ˜ν 1 , ˜ν 2 and ˜ν 3 is exactly the union **of** the three

paths.

Consider a set ˜C **of** pairwise parallel Jordan curves **of** the plane. These curves

split the plane into several block regions. Consider the set **of** all the block regions

bordered by some elements **of** ˜C. We are interested by the inclusion-minimal elements

**of** this set, that we call minimal block regions formed by ˜C. The following proposition

comes directly from the definition **of** the minimal block-regions:

Proposition 8. Let ˜C be a set **of** pairwise parallel curves in the planeΣ. A set **of**

pointsA**of** the plane are contained in a same minimal block-region formed by ˜C if

and only if for any ˜ν∈ ˜C, there is a one-block region (˜ν,R(˜ν)) containingA.

7 Minimal triangulations **of**G

LetGbe a 3-connected **planar** graph and letH be a minimal triangulation **of**

G. According to Theorem 1, there is a maximal set **of** pairwise parallel separators

Γ⊆∆ G such thatH =G Γ . LetC(Γ ) ={ν S |S∈Γ} be the cycles associated to

the minimal separators **of**Γ and let ˜C(Γ ) ={˜ν S |S∈Γ} be the curves associated

to these cycles. According to proposition 7, the cycles **of**C(Γ ) are pairwise parallel.

Thus, the curves **of** ˜C(Γ ) split the plane into block regions. We show that any

maximal cliqueΩ **of**H corresponds to the original vertices contained in a minimal

block region formed by ˜C(Γ ).

IfBR is a block region, we denote byBR G the vertices **of**Gcontained inBR.

Theorem 2. LetH =G Γ be a minimal triangulation **of** a 3-connected **planar** graph

G.Ω⊆V is a maximal clique **of**H if and only if there is a minimal block region

BR formed by ˜C(Γ ) such thatΩ =BR G .

Pro**of**. LetBR be a minimal block region formed by ˜C(Γ ), we show thatΩ=BR G

is a clique **of**H. Suppose there are two verticesx,y∈Ω, non adjacent inH. Thus,

there is a minimal separatorS **of**H separatingxandyinH. ThenS is also

a minimal separator **of**G, separatingxandyinG(cf. Theorem 1). Therefore,

˜ν S ∈ ˜C(Γ ) separatesxandy in the plane, contradicting proposition 8.

LetΩ be a clique **of**H. For any minimal separatorS **of**H there is a connected

componentC(S) **of**H\S such thatΩ⊆S∪C(S). By Theorem 1,S∈Γ andC(S)

is a connected component **of**G\S, so we deduce that the points **of**Ω are contained

in a same one-block region (˜ν S ,R(˜ν S )) defined by ˜ν S . This holds for eachS∈Γ ,

because the minimal separators **of**H are exactly the elements **of**Γ . We conclude by

proposition 8 thatΩ is contained in some minimal blockBR formed by ˜C(Γ ). ⊓⊔

8 Triangulations **of** the dual graphG ∗

LetGbe a plane graph andC be a set **of** pairwise parallel cycles **of**G I . The family

˜C **of** curves associated to these cycles splits the plane into block regions. LetG ∗ be

the dual **of**G. We show in this section how to associate toC a triangulationH(C)

**of**G ∗ such that each clique **of**H(C) corresponds to the face-vertices contained in

some minimal block-region defined by ˜C.

Definition 7. Consider a **planar** embedding **of** the graphG = (V,E) and letG ∗ =

(F,E ∗ ) the dual **of**G. LetC be a set **of** pairwise parallel cycles **of**G I . We define the

9

graphH(C) = (F,E H ) with vertex setF, we place an edge between two face-vertices

f andf ′ **of**H if and only iff andf ′ are in same a minimal block region defined

by ˜C.

Theorem 3.H(C) is a triangulation **of**G ∗ . Moreover, for any cliqueΩ ∗ **of**H(C)

there is some minimal block regionBR defined by ˜C such thatΩ ∗ is formed by the

face-vertices contained inBR.

Pro**of**. We show thatH is a supergraph **of**G ∗ . Letff ′ be an edge **of**G ∗ , clearly no

cycle **of**G I crosses the edgeff ′ in the plane. Thus, for any cycle ˜ν **of** ˜C, ˜ν does not

separate the pointsf andf ′ . By proposition 8,f andf ′ are in a same block region

formed by ˜C, s**of**f ′ is an edge **of**H(C).

We prove now thatH(C) is chordal. Suppose there is a chordless cycleν H **of**

H(C), having at least four vertices. Letf,f ′ be two non-adjacent vertices **of**ν H . By

proposition 8, there is a curve ˜ν∈ ˜C separating the pointsf andf ′ in the plane.

Consider the two interior-disjoint pathsµ 1 andµ 2 fromf t**of** ′ inν H . We show

that at least one face-vertex **of** each **of** these paths belongs to ˜ν. Letµ 1 = [f =

f 1 ,f 2 ,...,f p =f ′ ].

LetRandR ′ be the regions **of**Σ\˜ν containingf, respectivelyf ′ . Letf j the

last point **of**µ 1 contained inR, so 1≤j0. Letxandy two vertices **of**BR G , by induction hypothesis there

exists a cycleν ′ containingxandyinside∩ k i=2 ( ˜ν i,R( ˜ν i )). Ifν ′ intersectsν 1 in at

most one vertex then the cycleν ′ is insideBR.

10

Otherwise, letµ x (resp.µ y ) be the path **of**ν ′ that containsx(resp.y) and whose

the only vertices in common withν 1 are its endsx 1 andx 2 (resp.y 1 andy 2 ). If

µ x =µ y then we can completeµ x in a simple cycle that belongs to ( ˜ν 1 ,R( ˜ν 1 )) by

followingν 1 fromx 1 tox 2 . Ifµ x ≠µ y , on the cycleν 1 , the verticesx 1 andx 2 and

the verticesy 1 andy 2 are juxtaposed (see figure 3). There are two disjoint pathsµ 1

andµ 2 **of**ν 1 whose ends arex 1 andx 2 , respectivelyy 1 andy 2 . The four pathsµ 1 ,

µ 2 ,µ x andµ y form a simple cycle that lies inside (˜ν 1 ,R(˜ν 1 )).

x 1 y 1

µ 1

˜ν

x

y

µ 2

x 2 y 2

˜ν ′

Fig. 3. Cycle inside the block-region

Moreover in both cases, sinceν 1 also lies inside∩ n i=2 ( ˜ν i,R( ˜ν i )) the new cycle

lies inside∩ n i=2 ( ˜ν i,R( ˜ν i )) and so insideBR.

In any case, we can exhibit a cycle insideBR passing throughxandy soBR G

is 2-connected.

⊓⊔

Lemma 10. LeG = (V,E) be a 2-connected **planar** graph, consider a familyC **of**

pairwise parallel cycles. LetBR be a block-region defined by a subfamily ˜C ′ **of** ˜C and

ν∈C ′ .

Either all verticesBR G are onν or there exists a pathµ⊆BR G which intersects

ν only in its extremities.

Pro**of**. Suppose there exists a vertexx∈BR G which is not onν. We know by

Lemma 9 thatBR G is 2-connected. Take two verticesy andz onν, applying Dirac’s

fan lemma toxand{y,z}, we get two disjoint paths except inx,µ y andµ z inBR G ,

connecting respectivelyxtoyandxtoz. Cuttingµ y (resp.µ z ) at the first vertex

y ′ (resp.z ′ ) onν, we obtain two pathsµ y ′ andµ z ′ which intersectν only iny ′ and

z ′ . Then the concatenation **of**µ y ′ andµ z ′ is a suitable pathµ. ⊓⊔

Theorem 4. LetG = (V,E) be a 3-connected **planar** graph. LetC be an inclusion

maximal family **of** pairwise parallel cycles **of**G I . Consider a minimal block-region

BR **of**G I defined by ˜C then eitherBR GI =ν orBR GI =ν∪µ, whereν is inC

andµis a path which touchesν only in its ends.

Pro**of**. IfBR GI is not a cycle, we know by Lemma 10 that there exists a pathµ

insideBR GI that intersectsν only on its extremitiesxandy. The verticesxand

y define two subpathsν 1 andν 2 **of**ν, so we have three cycles contained inBR GI ,

namelyν,µν 1 andµν 2 . These cycles are pairwise parallel and parallel to the cycles

**of**C, so by maximality **of**C they are inC. These three cycles define a block-region

BR ′ which is exactly ˜ν∪ ˜µ. SinceBR ′ = ˜ν∪ ˜µ⊆BR andBR is a minimal blockregion

we can conclude thatBR ′ =BR. ⊓⊔

11

Theorem 5. LetG = (V,E) be a 3-connected **planar** graph without loops. Then

tw(G)−1≤tw(G ∗ )≤tw(G) + 1.

Pro**of**. By duality, it is sufficient to prove the second inequality. SinceGis 3-

connected without loops,G,G ∗ and, by proposition 1,G I are 2-connected without

loops. LetC be a family **of** cycles **of**G I that gives a triangulationH **of**Gwith

ω(H)−1 =tw(G). We completeC into a maximal familyC ′ **of** pairwise parallel

cycles **of**G I .

According to Theorem 3, the familyC ′ defines a triangulationH ∗ **of**G ∗ . Let

BR be a minimal block-region with respect to ˜C ′ . By Theorem 4, eitherBR GI =ν

orBR GI =ν∪µ. In the first case, sinceG I is bipartite we have|BR GI ∩V| =

|BR GI ∩V ∗ |. In the later case, the difference between the number **of** vertices **of**

G andG ∗ **of**BR GI comes fromµ. Once again, sinceG I is bipartite the difference

can be at most one. But each minimal block-region formed by ˜C ′ is contained in a

minimal block-region formed by ˜C, so, by Theorem 2,BR GI ∩V is a clique **of**H.

Therefore the maximal cardinality **of** a clique inH ∗ is the maximal cardinality **of** a

clique inH plus one and the second inequality is proved. ⊓⊔

10 Planar **graphs** which are not 3-connected

We have proved that, for any 3-connected **planar** graphG, the treewidth **of** its dual

is at most the treewidth **of**Gplus one. We extend this result to arbitrary **planar**

**graphs**.

The following lemma is a well-known result, see for example [12] for a pro**of**:

Lemma 11. LetG = (V,E) be a graph (not necessarily **planar**) andS be a separator

**of**Gsuch thatG[S] is a complete graph. LetV 1 ,V 2 ⊆V such thatS,V 1 and

V 2 form a partition **of**V andS separates each vertex **of**V 1 from each vertex **of**V 2 .

Then tw(G) = max(tw(G[S∪V 1 ]), tw(G[S∪V 2 ])).

Lemma 12. LetG=(V,E) be a graph, not necessarily **planar**. Suppose thatGhas

a minimal separatorS={x,y} **of** size two. LetG xy = (V,E∪{xy}) be the graph

obtained fromGby adding the edgexy. Then tw(G xy ) = tw(G).

Pro**of**. Ifxy is an edge **of**G, thenG xy =G. Suppose thatxy is not an edge **of**G.

SinceGis a minor **of**G xy , we have tw(G)≤tw(G xy ), so it remains to show that

tw(G)≥tw(G xy ).

S is also a minimal separator **of**G ′ , so letC be a full component associated to

S inGand letV 2 =V\ (C∪S). LetG 1 =G xy [S∪C] andG 2 =G[S∪V 2 ]. By

Lemma 12, we have tw(G xy ) = max(tw(G 1 ), tw(G 2 )). It is sufficient to prove that

tw(G 1 )≤tw(G) and tw(G 2 )≤tw(G). We show thatG 1 andG 2 are minors **of**G.

By Lemma 1, there is a full componentDassociated toS different fromC,

soD⊆V 2 . There is a pathµfromxtoyinG[D∪{x,y}], and the interior **of**

µ avoids the vertices **of**G 1 . Therefore,G 1 is a minor **of**G[S∪C∪µ], soG 1 is

a minor **of**S. In a similar way, there is a pathµ ′ fromxtoyinG[C∪{x,y}],

soG 2 is a minor **of**G[V 2 ∪S∪µ ′ ] and thus a minor **of**G. We conclude that

tw(G)≥max(tw(G 1 ), tw(G 2 )) = tw(G xy ). ⊓⊔

Lemma 13. Suppose there is a plane graphGnot satisfying tw(G ∗ )≤tw(G) + 1

and there is a separatorS ={x,y} **of**G. ThenG S also contradicts tw(G ∗ S )≤

tw(G S ) + 1.

Pro**of**. By Lemma 12, tw(G S ) = tw(G). We also have thatG S is **planar** andG ∗ is

a minor **of**G ∗ S . Indeed, ifxy is not an edge **of**G, letC be a full component **of**G\S

12

andν S (C) the cycle associated toS andC, close toC. Thenν S (C) = [x,f,y,f ′ ],

sox,y are incident to a same facef. We obtain a plane drawing **of**G S by adding

the edgexy in the facef. The new edge will split the facef into two facesf 1

andf 2 , and clearly the dual **of**Gis obtained from the dual **of**G S by contracting

the edgef 1 f 2 into a single vertexf. Therefore, tw(G ∗ )≤tw(G ∗ S ). Consequently, if

tw(G ∗ S )≤tw(G S) + 1, then tw(G ∗ )≤tw(G) + 1. ⊓⊔

Theorem 6. For any plane graphG,

tw(G ∗ )≤tw(G) + 1.

Pro**of**. Suppose there is a graphGsuch that tw(G ∗ )>tw(G) + 1. We takeGwith

minimum number **of** vertices. It is easy to check thatGmust have at least four

vertices.

By Theorem 5,Gis not 3-connected, so letS be a minimal separator **of**Gwith

at most two vertices. According to Lemma 13, we can consider thatS is a clique

inG. LetC be a connected component **of**G\S, we denoteG 1 =G[S∪C] and

G 2 =G[V\C] (ifGis not connected, thenS=∅ andC is a connected component

**of**G). By Lemma 11, tw(G) = max(tw(G 1 ), tw(G 2 )).

The **graphs**G 1 andG 2 are clearly **planar** and they have less vertices thatG, so

tw(G ∗ 1)≤tw(G 1 ) + 1 and tw(G ∗ 2)≤tw(G 2 ) + 1. It remains to prove that tw(G ∗ )≤

max(tw(G ∗ 1 ), tw(G∗ 2 )).

Consider the case whenGis 2-connected. By proposition 3 there is a cycle

ν S (C) **of**G I associated toS andC, close toC. The cycle contains four vertices,

˜ν S (C) = [x,f,y,f ′ ]. LetR 1 (respectivelyR 2 ) be the region **of**Σ\ν S (C) containing

C (respectivelyV\ (S∪C)). Notice that the vertices **of**G 1 (respectivelyG 2 ) are

exactly the original vertices **of** (˜ν S (C),R 1 ) (respectively (˜ν S (C),R 2 )). LetF 1 andF 2

be the face-vertices **of**G I contained inR 1 , respectivelyR 2 . We denoteS f ={f,f ′ }.

LetG f 1 be the graph obtained fromG∗ [S f ∪F 1 ] by adding the edgeff ′ . Consider

the plane drawing **of**G 1 obtained by restricting the drawing **of**Gat the one-block

region (˜ν S (C),R 1 ) and by adding the edgexy through the regionR 2 . It is easy to see

thatG f 1 is exactly the dual **of**G 1. In a similar way, we define the graphG f 2 obtained

fromG ∗ [S f ∪F 2 ] by adding the edgeff ′ . If we consider the plane drawing **of**G 2

obtained by restricting the drawing **of**Gto the one-block region (˜ν S (C),R 2 ) and by

adding the edgexy throughR 2 , thenG f 2 is the dual **of**G 2. By the minimality **of**G,

we have tw(G f 1 )≤tw(G 1) + 1 and tw(G f 2 )≤tw(G 2) + 1. Observe now that in the

graphG ∗ S f

obtained fromG ∗ by adding the edgexy,S f separatesF 1 fromF 2 . By

Lemma 11, tw(G ∗ S f

) = max(tw(G f 1 ), tw(G2 f )), so tw(G∗ S f

)≤max(tw(G 1 ), tw(G 2 ))+

1 = tw(G) + 1. We conclude that tw(G ∗ )≤tw(G) + 1.

The case whenGis not 2-connected is similar. Suppose thatGis connected

but no 2-connected, soS has a unique vertexx. There is a facef **of**G I such that

we can draw a Jordan curve ˜ν S passing throughxandf, contained in the facef

(except the pointx), and the curve separatesC fromV\ (C∪{x})). IfGis not

connected, we can take a connected componentC and a facef such that a Jordan

curve ˜ν S contained in the facef, passing throughf, separatesC fromV\C. As

in the case **of** 2-connected **graphs**, we consider the regionsR 1 (respectivelyR 2 ) **of**

Σ\˜ν S containingC (respectivelyV\ (C∪S)). We takeS f ={f} and we denote

F 1 (respectivelyF 2 ) the face-vertices **of**G I contained inR 1 (respectivelyR 2 ). Then

G f 1 =G∗ [S f ∪F 1 ] is the dual **of**G 1 andG f 2 =G∗ [S f ∪F 2 ] is the dual **of**G 2 .

We conclude that tw(G ∗ ) = max(tw(G f 1 ), tw(Gf 2 ))≤max(tw(G 1), tw(G 2 )) + 1, so

tw(G ∗ )≤tw(G) + 1.

⊓⊔

13

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